Essentials of Computational Chemistry

2.4 GEOMETRY OPTIMIZATION 47
that the speed of a force-field calculation exhibits with respect to increasing system size.
Although we raise the issues here in the context of geometry optimization, they are equally
important in force-field simulations, which are discussed in more detail in the next chapter.
If we look at the scaling behavior of the various terms in a typical force field, we see that
the internal coordinates have very favorable scaling – the number of internal coordinates
is 3N − 6, which is linear in N. The non-bonded terms, on the other hand, are computed
from pairwise interactions, and therefore scale as N 2 . However, this scaling assumes the
evaluation of all pairwise terms. If we consider the Lennard–Jones potential, its long-range
behavior decays proportional to r −6 . The total number of interactions should grow at most
as r 2 (i.e., proportional to the surface area of a surrounding sphere), so the net energetic
contribution should decay with an r −4 dependence. This quickly becomes negligible (particularly
from a gradient standpoint) so force fields usually employ a ‘cut-off’ range for the
evaluation of van der Waals energies – a typical choice is 10 ˚A. Thus, part of the calculation
involves the periodic updating of a ‘pair list’, which is a list of all atoms for which the
Lennard–Jones interaction needs to be calculated (Petrella et al. 2003). The update usually
occurs only once every several steps, since, of course, evaluation of interatomic distances
also formally scales as N 2 .
In practice, even though the use of a cut-off introduces only small disparities in the energy,
the discontinuity of these disparities can cause problems for optimizers. A more stable
approach is to use a ‘switching function’ which multiplies the van der Waals interaction
and causes it (and possibly its first and second derivatives) to go smoothly to zero at some
cut-off distance. This function must, of course, be equal to 1 at short distances.
The electrostatic interaction is more problematic. For point charges, the interaction energy
decays as r −1 . As already noted, the number of interactions increases by up to r 2 ,sothe
total energy in an infinite system might be expected to diverge! Such formal divergence is
avoided in most real cases, however, because in systems that are electrically neutral there
are as many positive interactions as negative, and thus there are large cancellation effects. If
we imagine a system composed entirely of neutral groups (e.g., functional groups of a single
molecule or individual molecules of a condensed phase), the long-range interaction between
groups is a dipole–dipole interaction, which decays as r −3 , and the total energy contribution
should decay as r −1 . Again, the actual situation is more favorable because of positive and
negative cancellation effects, but the much slower decay of the electrostatic interaction makes
it significantly harder to deal with. Cut-off distances (again, ideally implemented with smooth
switching functions) must be quite large to avoid structural artifacts (e.g., atoms having large
partial charges of like sign anomalously segregating at interatomic distances just in excess
of the cut-off).
In infinite periodic systems, an attractive alternative to the use of a cut-off distance is the
Ewald sum technique, first described for chemical systems by York, Darden and Pedersen
(1993). By using a reciprocal-space technique to evaluate long-range contributions, the total
electrostatic interaction can be calculated to a pre-selected level of accuracy (i.e., the Ewald
sum limit is exact) with a scaling that, in the most favorable case (called ‘Particle-mesh
Ewald’, or PME), is NlogN. Prior to the introduction of Ewald sums, the modeling of
polyelectrolytes (e.g., DNA) was rarely successful because of the instabilities introduced